We can start by simplifying each factor separately and then combine them.
(a + b)/(a^2 + b^2) can be simplified by multiplying both the numerator and denominator by (a - b):
(a + b)/(a^2 + b^2) * (a - b)/(a - b) = (a^2 - b^2)/(a^3 - b^3)
Next, we simplify a/(a - b) by multiplying both the numerator and denominator by (a + b):
a/(a - b) * (a + b)/(a + b) = a(a + b)/(a^2 - b^2)
Lastly, we simplify (a^4 - b^4)/(a + b)^2 by factoring the numerator and expanding the denominator:
(a^4 - b^4)/(a + b)^2 = [(a^2)^2 - (b^2)^2]/(a + b)^2 = [(a^2 + b^2)(a^2 - b^2)]/(a + b)^2
Now we can combine all three simplified factors:
(a + b)/(a^2 + b^2) * a/(a - b) * (a^4 - b^4)/(a + b)^2 = [(a^2 - b^2)/(a^3 - b^3)] * [a(a + b)/(a^2 - b^2)] * [(a^2 + b^2)(a^2 - b^2)]/(a + b)^2
Simplifying further, we can cancel out the (a^2 - b^2) terms and the (a + b) terms:
= [a(a + b)/(a^3 - b^3)] * [(a^2 + b^2)/(a + b)]
= a(a + b)(a^2 + b^2)/(a + b)(a^3 - b^3)
= a(a^2 + b^2)/(a^3 - b^3)
Therefore, the simplified expression is a(a^2 + b^2)/(a^3 - b^3)